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Clicker Question 1

Clicker Question 1. What is the general solution to the DE x y  = x y – y ln( x ) ? A. y = x – (1/2) (ln( x )) 2 + C B. y = A e ( x – (1/2) (ln( x ))^2) (A > 0) C. y = A e ( x – ln(ln( x ))) (A > 0) D. y = e ( x – (1/2) (ln( x ))^2) + C

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Clicker Question 1

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  1. Clicker Question 1 • What is the general solution to the DEx y = xy – y ln(x) ? • A. y = x – (1/2)(ln(x))2 + C • B. y = A e(x – (1/2)(ln(x))^2) (A > 0) • C. y = A e(x – ln(ln(x))) (A > 0) • D. y = e(x – (1/2)(ln(x))^2) + C • E. y = e(x – ln(ln(x))) + C

  2. Clicker Question 2 • What function y satisfies that its rate of change is always equal to cot(y) and y = /3 when x = 0. • A. y = arcsec(2ex) • B. y = arcsin(2x) • C. y = sec(x + /3) • D. y = arcsec((/3)ex) • E. y = arcsec(ex + 2)

  3. Application of DE’s: Population Growth (3/26/14) • Let P be the size of a population and let t be time. For example, if the population grows at a rate proportional to its size, this say that it satisfies the DE: dP / dt = kP , k being the relative growth rate. • This is separable, and we know the general solution is P = A e kt where A is the starting population. • This is, naturally, called exponential growth.

  4. The Logistic Model of Growth • Many populations may grow exponentially at first, but eventually that growth rate slows as capacity (space, food, etc.) is reached. • That is, as time passes, k will approach 0. • If the maximum capacity of the population is denoted M, a simple expression which approaches 0 as P approaches M is 1 – P / M .

  5. The Logistic DE • Thus a DE which would model this “exponential growth at first but slowing of the growth rate as P approaches its maximum capacity” would be

  6. Example • Suppose a population growing by the logistic model has a maximum capacity of 1000 and displays an initial growth rate of 8%. • Look at an Euler’s Method approximate solution assuming an initial population of 2. • Can we explicitly solve this DE? Is it separable?

  7. Assignment for Friday • Hand-in #3 is due at class time. • Read Section 9.4 through page 608. • The logistic DE is separable. Separate it. Can you now solve it?

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